Orbifolds and groupoids (Q409514)

The authors take the point of view, after a construction due to [\textit{D. Pronk}, Groupoid representation for sheaves on orbifolds. PhD Thesis. Utrecht (1995)], of smooth complex orbifolds as groupoid objects. Specifically, this construction allows for every smooth reduced orbifold atlas to be associated to a proper étale and effective groupoid object over smooth manifolds. The authors define a 2-category structure, denoted (Pre-Orb), on the category of reduced complex orbifold atlases, construct a 2-functor \(F\) from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds, and prove that \(F\) induces a bijection between the two natural equivalence classes (equivalent orbifold atlases and Morita equivalent groupoid objects, respectively) of (Pre-Orb) and (Grp).